The objectives of this proposal are to carry out statistical research on monitoring, design and analysis of clinical trials using sequential methods for early termination, and to investigate the implications of early termination in the interpretation of the results from such clinical trials. We will focus primarily on Phase III clinical trials in which the major outcome variable of interest is a failure time. There seems to be a consensus that a clinical trial should be monitored throughout the study duration and be terminated on grounds of ethics and efficacy if there is substantial evidence of treatment difference or lack of such difference. The objective of this proposal is to propose stochastically curtailed tests based on conditional or predictive power for early termination of clinical trials which have initially been designed as a fixed duration study, to investigate the relationship between the stochastic curtailment tests and the group sequential tests, and to explore the consequence of the use function approach during sequential design on the operating characteristics. When clinical trials are sequentially monitored for possible early termination, it reduces the sensitivity of tests employed to detect a certain treatment difference as a consequence of adjustment necessary to maintain the significance level as a consequence of repeated significance testing. Hence the "sample size" or the study duration should be adequately adjusted so that the desired sensitivity is maintained. Specifically, we propose to investigate the asymptotic property of sequentially computed stratified log rank test statistics over time, to develop methods for determining the study duration for clinical trials with (i) stratified randomization and (ii) multiple endpoints. The ordering of the sample space in sequential analysis dictates the interpretation of the observed significance level, the so-called p value. The usual maximum likelihood estimators in sequential experiments are known to be biased due to the optional sampling. As a result there is a lot of confusion and controversy over the role and the interpretation of repeated significance testing. Hence we would like to review and compare properties of different methods for constructing confidence intervals following sequential tests and of various orderings of the sample space in sequential analysis. Also we propose to investigate the point and confidence interval estimation procedure for the secondary endpoint following early stopping based on the primary endpoint in sequentially conducted clinical trails.